Engineering Letters, 19:3, EL_19_3_17
______________________________________________________________________________________
Harmonics and Interharmonics Measurement
using Group-harmonic Energy Distribution
Minimizing Algorithm
Hsiung Cheng Lin, Chao Hung Chen, Liang Yih Liu
Abstract—The Fast Fourier Transform (FFT) is still a
widely-used tool for analyzing and measuring both stationary
and transient signals in power system harmonics. However, the
misapplications of FFT can lead to incorrect results caused by
some problems such as aliasing effect, spectral leakage and
picket-fence effect. A strategy of iterative Group-harmonic
Energy Distribution Minimizing (GEDM) algorithm is
developed for system-wide harmonic/inter-harmonic evaluation
in power systems. The proposed algorithm can restore the
dispersing spectral leakage energy caused by the FFT, and
regain its harmonic/inter-harmonic magnitude and respective
frequency. Every iteration loop for harmonic/interharmonic
evaluation can guarantee to be convergent. Consequently, not
only high-precision in integer harmonic measurement can be
retained, but also the inter-harmonics can be identified
accurately, particularly under system frequency drift. The
numerical example is presented to verify the proposed algorithm
in term of robust, fast and precise performance.
Index Terms— harmonics, inter-harmonics, group-harmonics,
DFT, FFT
I. INTRODUCTION
W
ITH increasing use of power electronic systems and
time-variant non-linear loads in industry, the generated
power harmonics and interharmonics have resulted in
serious power line pollution. Power supply quality is
therefore aggravated. Traditional harmonics may cause
negative effects such as signal interference, overvoltage, data
loss, equipment malfunction, equipment heating and damage,
etc. The noise on data transmission line is also related with
harmonics. At some special systems, harmonic current
components may cause effect of carrier signals, and thus
interfere other carrier signals. As a result, some facilities may
be affected. Once harmonics source enter computer
instruments, the data stored in the computer may be lost up to
ten times. Moreover, harmonics may also cause transformer
and capacitor over heating, thus reducing their working life.
The resulting rotor heating and pulsating output torque will
decrease the driver’s efficiency [1-8].
Manuscript received Nov. 25, 2010; revised Feb. 17, 2011. This work
was supported by the National Science Council of the Republic of China,
Taiwan under Grant No. NSC 99-2221-E-167 -034.
Hsiung Cheng Lin is with the Department of Electronic Engineering,
National Chin-Yi University of Technology, Taiwan (e-mail:
hclin@ncut.edu.tw).
Chao Hung Chen is with the Department of Automation Engineering,
Chienkuo
Technology
University,
Taiwan
(e-mail:
jauhorng.chen@gmail.com).
Liang Yih Liu is with the Department of Automation Engineering,
Chienkuo Technology University, Taiwan (e-mail: lliu@cc.ctu.edu.tw).
The presence of power system interharmonics has not only
brought many problems as harmonics but produced additional
problems. For instance, there are thermal effects, low
frequency oscillation of mechanical system, light and CRT
flicker, interference of control and protection signals, high
frequency
overload
of
passive
parallel
filter,
telecommunication interference, acoustic disturbance,
saturation of current transformer, subsynchronous
oscillatoions, voltage fluctuations, malfunctioning of remote
control system, erroneous firing of thyristor apparatus, and
the loss of useful life of induction motors, etc. These
phenomenons may even happen under low amplitude [1-4].
Conventionally, Discrete Fourier transform (DFT) method
is efficient for signal spectrum evaluation because of the
simplicity and easy implementation. The use of the FFT can
reduce the computational time required for DFT by several
orders of magnitude. An improper use of DFT (or FFT) based
algorithms can, however, lead to multiple interpretations of
spectrum [4-6]. For example, if the periodicity of DFT data
set does not match the periodicity of signal waveforms, the
spectral leakage and picket-fence effect will occur. Since the
power system frequency is subject to small random deviations,
some degree of spectral leakage can not be avoided. A
number of algorithms, e.g., short time Fourier Transform [7],
least-square approach [8-10], Kalman filtering [11-12],
artificial neural networks [6,13], have been proposed to
extract harmonics. The approaches may either suffer from low
solution accuracy or less computational efficiency. None is
reported to perform well in interharmonic identification under
system frequency variations though each demonstrates its
specific advantages.
The presence of interharmonics strongly poses difficulties
in modeling and measuring the distorted waveforms. This is
mainly due to: 1) very low values of interests of
interharmonics (about one order of quantity less than for
harmonics), 2) the variability of their frequencies and
amplitudes, 3) the variability of the waveform periodicity, and
4) the great sensitivity to the spectral leakage phenomenon. In
recent years, the effect caused by interharmonics is being
worsened apparently. Therefore, now the development of
accurate interharmonics measurement has attracted great
attention both industry and academics. This point of view is
fully supported by exploring a number of publications
(2007-2011) related to this field [14-36]. However, the
published outcome may still suffer from low accuracy, long
computational time, complexity or measurement limitation,
etc. Accordingly, it is still an essential research issue to be
carried on in this field.
IEC 61000-4-7 established a well disciplined measurement
method for harmonics/interharmonics. This standard recently
(Advance online publication: 24 August 2011)
Engineering Letters, 19:3, EL_19_3_17
______________________________________________________________________________________
has been revised to add methodology for measuring
inter-harmonics [37]. The key to the measurement of both
harmonics and inter-harmonics in the standard is the
utilization of a 10 or 12 cycle sample window upon which to
perform the Fourier transform. However, the spectrum
resolution with 5 Hz is not sufficiently precise to reflect the
practical inter-harmonic locations for both 50 Hz and 60 Hz
systems. This paper presents harmonic/inter-harmonic
identification using FFT-based GEDM approach which
retains the merits of FFT analysis and extends to
inter-harmonic identification under system frequency
variation environments. This paper is organized as follows.
Section II gives a background of the concept of system
harmonic/interharmonic measurement. Section III presents
the proposed GEDM algorithm. In Section IV, the model
validation with a numerical example is demonstrated.
Performance results under system frequency drift is included
and discussed. Conclusions are given in Section V.
I s* [ f k ] = P * [ f k ]
(5)
An interesting way to view this phenomenon is to observe
the FFT implementation, shown in Fig.1. Most leakages can
be collected into one group and are considered as though they
were all at the dominant harmonic frequency. The amplitude
of inter-harmonics (and/or sub-harmonics) can be thus
identified.
II. BACKGROUND OF SYSTEM HARMONIC/INTERHARMONIC
MEASUREMENT
The measurement of inter-harmonics is difficult with
results depending on many factors. Based on the so-called
“group” suggested by IEC 61000-4-7, the concept of
group-harmonic is introduced as follows [32].
By the Parseval relation in its discrete form, the power of
the waveform, P, can be expressed as
N / 2 −1
1 N / 2−1
(1)
P=
i s [ n] 2 = ∑ I s [ k ] 2
∑
N n=0
k =0
Both positive and negative values of spectral components
are considered to transform the frequency dominant sampled
signal into a periodic time dominant signal. Therefore, actual
signals spectral components relevant to symmetrical
frequencies are complex conjugates each other. However,
most real-world frequency analysis instruments display only
the positive half of the frequency spectrum because the
spectrum of a real-world signal is symmetrical around DC.
Thus, the negative frequency information is redundant.
For this reason, the power at the discrete frequency f k can
be expressed as
(2)
P[ f k ] = I s [k ] 2 + I s [ N − k ] 2 = 2 I s [k ] 2
where k=0,1, 2,…,N/2-1.
The RMS value of the harmonic amplitude at the discrete
frequency f k is
(3)
I h [ f k ] = P[ f k ] = 2 I s [k ]
The power of the harmonic at f k may disperse over a
frequency band around the f k due to the spectral leakage.
Hence, the total power of harmonics within the adjacent
frequencies around f k can be restored into a “group power”
*
[5]. Each “group power”, i.e., P [ f k ] , can be collected
between
f k − ∆k and f k + ∆k as follows.
+τ
P*[ f k ] =
∑ I h [ f k +∆k ]
2
(4)
∆k = −τ
where τ is an integer number and denotes the group
bandwidth.
Consequently, each harmonic amplitude can be estimated
as
Fig. 1 IEC subgrouping of “bins” for both harmonics and
interharmonics (graph reproduced from [3])
III. THE PROPOSED ITERATIVE GROUP-HARMONIC ENERGY
DISTRIBUTION MINIMIZING ALGORITHM
The power line waveform s (t ) (voltage/current) is
sampled using the sampling rate f s ( = 1 ) , which has the
Ts
fundamental frequency f d , as follows.
(6)
s(n) = s(t ) t =nTs , n = 0,1,2,...., N − 1
where N is the sampled point of Fourier fundamental period
Tf .
In general, the distorted signal can be composed of three
parts, as follows.
(7)
s(n) = sd (n) + sh (n) + si (n)
where s d (n) is the fundamental component, sh (n) is the
harmonic
components,
and
si (n)
represents
the
interharmonic components.
III.1 The Group-Harmonic Bin Power Algorithm
The length of the sampled window for FFT analysis plays
the critical point to determinate if the spectrum can be
achieved accurately. Based on the empirical observation
using FFT, the second stronger amplitude is found to be
located at the right side of the dominant component,
i.e., I h [ f k +1 ] > I h [ f k −1 ] , in case of
overlong
truncated-window. On the contrary, the second stronger
amplitude is located at the left side of the dominant
component, i.e., I h [ f k +1 ] < I h [ f k −1 ] , the truncated-window
length is insufficient for FFT analysis. Accordingly, the
proposed GEDM approach is to develop the mechanism for
correcting the window length according to the situation on the
dispersed energy. This proposed GEDM method in deed
extends the “group” concept that has been mentioned by IEC
61000-4-7 and some papers [3, 5, 20, 36-37].
(Advance online publication: 24 August 2011)
Engineering Letters, 19:3, EL_19_3_17
______________________________________________________________________________________
(5) Check if P ** [ f k ] ≤ Pmin . If yes, the iteration loop stops
'
and determine N = N ' . The fundamental frequency f d
Amplitude
and amplitude Ad' can be obtained. Otherwise, go back
to Step (2) to repeat the procedure until P ** [ f k ] ≤ Pmin .
Note that Pmin is a predefined minima power value.
III.2 The proposed GEDM Algorithm
The proposed Iterative Group-harmonic Energy
Distribution Minimizing (GEDM) algorithm that integrated
with the GBP algorithm is demonstrated as follows.
Frequency (Hz)
f d'
Fig. 2 Amplitude distribution around the dominant
component
Based on the above concept, the Group-Harmonic Bin
Power (GBP) algorithm is proposed and described as follows.
Ad'f d'
∆f '
s (n)
'
s (t )
f s = 5kHz, N = 1000
sd' (n) + sh' (n)
si' (n) = s ' (n) - [ sd' (n) + sh' ( n)]
I h [ f k +1 ] > I h [ f k −1 ]
I h [ f k +1 ] < I h [ f k -1 ]
P ** [ f k ] ≤ Pmin
Fig. 4 Flowchart of the proposed GEDM algorithm
f d'
Ad'
(1) Determine the new ∆f ' = f s using the GBP method and
N'
find the correct fundamental frequency f d' and its
respective amplitude Ad' . Accordingly, the fundamental
Fig. 3 The flowchart of the proposed GBP Algorithm
(1) Set f s = 5kHz ，N=1000 for sampling the power line
signal.
(2) Implement FFT.
(3) If I h [ f k +1 ] > I h [ f k −1 ] , N=N-1. Otherwise, go to next
step.
(4) If I h [ f k +1 ] < I h [ f k −1 ] , N=N+1. Otherwise, go to next
step.
frequency signal sd' ( n) and its harmonic signals sh' ( n )
can be obtained, as follows.
s ' (n) = s(t )
t=
n
= sd' (n) + sh' (n) + si' (n), n = 0,1,2,3,..., N ' −1
(8)
f s'
'
'
(2) Reconstruct the sd (n) and sh ( n) and form a
composed waveform. Therefore, the new waveform that
only contains interharmonic components without sd' ( n )
and sh' (n) can be obtained as follows.
(Advance online publication: 24 August 2011)
Engineering Letters, 19:3, EL_19_3_17
______________________________________________________________________________________
(9)
si' (n) = s ' (n) - [sd' (n) + sh' (n)]
(3) Assume the major interharmonic component (biggest
amplitude) as the fundamental component.
(4) Repeat steps (1) to (3) until all major interharmonics are
regained.
IV. MODEL VALIDATION WITH A NUMERICAL EXAMPLE
The proposed GEDM algorithm has been tested by the
synthesized line signal (voltage/current) to verify the
effectiveness of harmonic/inter-harmonic analysis. The
following example is used to illustrate the harmonic analysis
of a distorted waveform.
s( t ) = sin( 2πf d t - 35 ° ) + 0.3 sin( 2π ⋅ 3 ⋅ f d ⋅ t - 51 ° ) + 0 .25 sin( 2π ⋅ 5 ⋅ f d ⋅ t + 48 ° )
+ 0.2 sin( 2π ⋅ 131 ⋅ t + 12 ° ) + 0.25 sin( 2π ⋅ 213.5 ⋅ t - 75 ° ) + 0.1 sin( 2π ⋅ 351 ⋅ t )
Step (a): Measurement of fundamental and integer harmonics
with a 0.29 Hz frequency drift
In this case, the fundamental frequency component
including 3rd harmonic and 5th harmonic is considered to
have a 0.29 Hz variation. The dispersed power of the
harmonics over around the frequency band is significantly
reduced from 0.0088 to about zero within only 6 iteration
loops, shown in Fig. 7. Fig.8 indicates that each harmonic is
approaching toward its true amplitude step by step. The
amplitudes of fundamental, third and fifth component are thus
obtained as 1.0, 0.3 and 0.24 at the sixth iteration loop from
0.99, 0.28 and 0.22 at the first iteration loop, respectively.
Also, the fundamental frequency is found as 60.29 Hz,
matching the true one.
(10)
0.01
where f d = 60.29 Hz is the fundamental frequency.
Generally, the system frequency drift is a concern in power
systems because it may vary slightly from time to time due to
the change of system loads. This effect, in deed, influences the
traditional FFT spectrum analysis. As above, the line signal
has a fundamental frequency, i.e., 60.29 Hz, with 0.29 Hz
drift and a scaled amplitude of 1V. The 3rd and 5th harmonic
components are included in the synthesized waveform to
present a possible distorted waveform situation. Non-integer
components, i.e., interharmonic, such as 131 Hz, 213.5 Hz,
and 351 Hz are to be considered, reflecting a possible
polluted line case. Note that above harmonics/interharmonics
are assigned different magnitudes and phases.
According to the equation (10), we set f s = 5 kHz,
0.009
0.008
Dispersed Power
0.007
0.006
0.005
0.004
0.003
0.002
0.001
0
1
2
3
4
Iteration Number
5
6
Fig. 7 Convergent curve of the dispersed power at the
harmonic components
N = 1000 , i.e., ∆f = 5 Hz, and the waveform is shown in Fig.
5. As can be seen in Fig. 6, a considerable spectrum leakage
occurs using FFT so that the result is unable to represent its
actual spectrum.
1.2
1
Fundamental Harmonic(60.29Hz)
0.8
Amplitude
Third Harmonic(180.87Hz)
Fifth Harmonic(301.45Hz)
0.6
0.4
0.2
0
1
2
3
4
5
6
Iteration Number
Fig. 8 Amplitude tracking curve of the harmonic components
Fig. 5 The distorted waveform
Step (b): Measurement of the interharmonic at 213.5 Hz
In this stage, all harmonic components acquired at Step (a)
are excluded in the new waveform so that the interharmonic at
213.5 Hz is assumed as the fundamental component. The
dispersed power of the supposed fundamental band
(interharmonic at 213.5 Hz) is considerably reduced from
0.013 to almost zero within 8 iteration loops, shown in Fig. 9.
Accordingly, its amplitude is obtained as 0.25 from 0.21 and
the 213.5 Hz component is thus confirmed, shown in Fig. 10.
Fig. 6 Spectrum of the distorted waveform using FFT
The following steps are illustrated to find the true
harmonics/interharmonics.
(Advance online publication: 24 August 2011)
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0.014
0.3
0.012
0.25
0.2
Amplitude
Dispersed Power
0.01
0.008
0.006
0.15
0.1
0.004
0.05
0.002
0
0
1
1
2
3
4
5
6
7
Iteration Number
3
4
5
6
7
8
9
Iteration Number
Fig. 12 Amplitude tracking curve of the 131 Hz interharmonic
Fig. 9 Convergent curve of the dispersed power at the 213.5
Hz interharmonic
0.26
0.25
0.24
Dispersed Power
2
8
0.23
0.22
Step (d): Measurement of the interharmonic at 351 Hz
In the last stage, all harmonics, 213.5 Hz and 131
interharmonic are excluded in the new waveform. Therefore,
only 1 interharmonic (351 Hz) is remained. The dispersed
power of the supposed fundamental band (interharmonic at
351 Hz) is going down quickly to almost zero from 0.0088
within only 4 iteration loops, shown in Fig. 13. As a result, its
amplitude is obtained as 0.1 from 0.093 and the 351 Hz
component is thus confirmed, shown in Fig. 14.
0.21
0.01
0.009
0.2
0.008
0.19
2
3
4
5
6
7
8
Iteration Number
Fig. 10 Amplitude tracking curve of the 213.5 Hz
interharmonic
0.007
Dispersed Power
1
0.006
0.005
0.004
0.003
Step (c): Measurement of the interharmonic at 131 Hz
In this stage, all harmonics and 213.5 Hz interharmonic are
excluded in the new waveform. Similarly, the dispersed
power of the supposed fundamental band (interharmonic at
131 Hz) is approaching toward to zero from 0.0039 within 9
iteration loops, shown in Fig. 11. Accordingly, its amplitude
is obtained as 0.2 from 0.19 and the 131 Hz component is
therefore confirmed, shown in Fig. 12.
0.002
0.001
0
1
2
3
4
Iteration Number
Fig. 13 Convergent curve of the dispersed power at the 351
Hz interharmonic
0.2
0.18
0.16
0.004
0.14
0.0035
0.12
Amplitude
Dispersed Power
0.0045
0.003
0.0025
0.1
0.08
0.06
0.002
0.04
0.0015
0.02
0.001
0
1
0.0005
2
3
4
Iteration Number
0
1
2
3
4
5
6
7
8
9
Fig. 14 Amplitude tracking curve of the 351 Hz interharmonic
Iternation Number
Fig. 11 Convergent curve of the dispersed power at the 131
Hz interharmonic
V. CONCLUSIONS
Although the DFT (or FFT) has certain limitations in the
harmonic analysis, it is still widely used in industry today. The
harmonic/inter-harmonic identification using FFT-based
GEDM algorithm has been developed to be extracted
accurately and efficiently. The test results confirm that the
proposed GEDM method can guarantee the tracking of each
harmonic/interharmonic amplitude to be convergent at every
(Advance online publication: 24 August 2011)
Engineering Letters, 19:3, EL_19_3_17
______________________________________________________________________________________
iteration loop by the GBP algorithm. There is no theoretical
restriction in the locations of inter-harmonic components
while the group bandwidth ( τ ) of each
harmonic/inter-harmonic should be chosen appropriately.
Moreover, the GEDM methodology has been implemented
successfully by a LabVIEW programming so that it can be
easily extended to other software packages like
microprocessor for on-line measurement. Additionally, the
proposed GEDM can provide an advanced improvement for
most measurement devices with some inherent errors because
of
the
spectrum
leakages
caused
by
harmonics/inter-harmonics.
[18]
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(Advance online publication: 24 August 2011)